[Math] Minimal value of $\sum_{1\leq i

Question

Let $a_i \in \{-1,1\}$ for all $i=1,2,3,…,2014$ and $$M=\sum^{}_{1\leq i<j\leq 2014}a_{i}a_{j}.$$ Find the least possible positive value of $M$.

Came across this question in a Math Olympiad and I'm not sure how to even start, the answer given is 51.

Answer 1

Hint: $\displaystyle \left(\sum_{i=1}^{2014}a_i\right)^2=\sum_{i=1}^{2014}a_i\sum_{j=1}^{2014}a_j = \sum_{i=1}^{2014}a_i^2+2\sum_{1\leq i<j\leq 2014}a_i a_j.$